Unreferenced stub auto yes date December 2009 Disputeabout anyone ever having proposed scalarprogramming date March 2008 Programming paradigms Scalarprogramming is a term used to refer to those programming approaches that do not follow the array programmingprogramming paradigm paradigm . The fundamental idea behind scalarprogramming is that operations apply to a single value at a time. This makes it a low level programming model as it restricts the programmer to think and operate on the elemental constituents of data aggregates, and resorting to explicit loops of individual scalar operations to process the whole aggregates. See also Array programming contrast Programming paradigm s DEFAULTSORT ScalarProgramming Category Programming paradigms Comp sci stub ... more details
wiktionarypar scalarScalar may refer to Scalar mathematics , a quantity that can multiply vectors in the context of vector spaces Scalar physics , a quantity which is independent of specific classes of coordinate systems Scalar computing , an atomic quantity that can hold only one value at a time See also Scalar product, also known as the dot product Inner product space Scalar field Pseudoscalar Scalar prostoleg y Scalar processor Biology Pterophyllum P. scalare Pterophyllum scalare Lichtenstein, 1823 , a species of freshwater angelfish disambig bs Skalar vor de Skalar es Escalar eo Skalaro fr Scalaire he ms Skalar pl Skalar ro Scalar sr sh Skalar ... more details
Unreferenced stub auto yes date December 2009 Dablink See the scalar disambiguation scalar disambiguation page for an account of the broader concept used in physics and mathematics. In computing, a scalar is any non Object composition composite Value computer science value . Generally, all basic primitive data type s are considered scalar The boolean data type code bool code Numeric types code Integer computer science int code , the floating point types code float code and code double code Character types code Character computer science char char code and code String computer science string code The origin of the term scalar in computing dates back to the 1970 80s with the scalar processor , distinguishing it from a vector processor . See also Primitive data type Scalar processor DEFAULTSORT Scalar Computing Category Data types Datastructure stub ca Escalar inform tica de Skalare Variable es Escalar inform tica ... more details
Refimprove date December 2009 A scalar boson is a boson whose spin physics spin equals zero. Boson means that it has an integer valued Spin physics spin the scalar fixes this value to 0. The name scalar boson arises from quantum field theory . It refers to the particular transformation properties under Lorentz transformation . Examples Various known composite particles are scalar bosons, e.g. the alpha particle and the pi meson . Among the scalar mesons, one distinguishes between the Scalar meson scalar and Pseudoscalar meson pseudoscalar mesons, which refers to their transformation property under Parity physics parity . The only fundamental scalar boson in the standard model of elementary particle physics is the Higgs boson . It is the only elementary particle in the Standard Model that has not yet been experimentally measured February 2012 . There are various other hypothetical fundamental scalar bosons, including the inflaton . One very popular quantum field theory, which uses scalar bosonic fields and is introduced in many introductory books to quantum field theories ref cite book author Michael E. Peskin and Daniel V. Schroeder title An Introduction to Quantum Field Theory publisher Westview Press year 1995 isbn 0 201 50397 2 ref for pedagogical reasons, is the so called Quartic interaction math Phi 4 math theory . It usually serves as a toy model to introduce into the basic concepts of the field. See also Scalar meson Pseudoscalar meson Quantum field theory Scalar field theory Vector boson References reflist Particle stub date January 2012 DEFAULTSORT Scalar Boson Category Bosons Category Mesons Category Quantum field theory ... more details
File Scalarfield.jpg thumb right A scalar field such as temperature or pressure, where intensity of the field is graphically represented by intensity of the color. In mathematics and physics , a scalar field associates a scalar value to every point in a space. The scalar may either be a scalar mathematics mathematical number , or a scalar physics physical quantity . Scalar fields are required to be coordinate ... of the scalar field at the same point in space or spacetime . Examples used in physics include the temperature ... fields, such as the Higgs boson Higgs field . These fields are the subject of scalar field theory . Definition Mathematically, a scalar field on a region U is a real valued function real or complex ... title Scalar id s s083240 ref The region U may be a set in some Euclidean space , Minkowski space ... to some order. A scalar field is a tensor field of order zero, ref springer id s s083260 title Scalar field ref and the term scalar field may be used to distinguish a function of this kind with a more general tensor field, density bundle density , or differential form . Physically, a scalar ..., a scalar field should also be independent of the coordinate system used to describe the physical ... on the numerical value of a scalar field at any given point of physical space. Scalar fields are contrasted ..., scalar fields are often contrasted with pseudoscalar fields. Uses in physics In physics, scalar fields ... , which can be obtained as the gradient of the potential energy scalar field. Examples include ... , are scalar fields which describe the more familiar forces. A temperature , humidity or pressure ... field theory , a Bosonic field scalar field is associated with spin 0 particles. The scalar field may be real or complex valued. Complex scalar fields represent charged particles. These include the charged ... In the Standard Model of elementary particles, a scalar Higgs field is used to give the lepton s and massive ... . In scalar theories of gravitation scalar fields are used to describe the gravitational field ... more details
Dablink See scalar disambiguation scalar for an account of the broader concept also used in mathematics and computer science. In physics , a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations in Newtonian mechanics , or by Lorentz transformation s or space time translations in relativity . This is in contrast to a vector physics vector . A related concept is a pseudoscalar physics pseudoscalar , which is invariant under proper rotation s but like a pseudovector flips sign under improper rotation s. Physical quantity Main Physical quantity A physical ..., other changes of the coordinate system may affect the formula for computing the scalar for example, the Euclidean formula for distance in terms of coordinates relies on the basis being orthonormal , but not the scalar ... dimensional space is a scalar, but the Direction geometry, geography direction from one of those points ... on the horizontal plane and the angle away from that plane. Force cannot be described using a scalar ... of a force alone can be described with a scalar, for instance the gravitation al force acting on a particle is not a scalar, but its magnitude is. The speed of an object is a scalar e.g. 180 km h , while its velocity is not i.e. 180 km h north . Examples of scalar quantities in Newtonian mechanics electric charge and charge density An example of a pseudoscalar is the scalar triple product ... physically . Scalars in relativity theory Main Lorentz scalar In the theory of relativity , one considers ... density at a point in a medium, which is a scalar in classical physics, must be combined with the local ... be combined with momentum density and pressure into the stress energy tensor . Examples of scalar quantities ..., density are further examples of scalar quantities. See also Scalar field Scalar field theory Scalar mathematics Relative scalar Euclidean vector Notes Reflist 3 References cite book last Arfken first ... 9045 6 DEFAULTSORT Scalar Physics Category Introductory physics Category Fundamental physics concepts ... more details
The Scalar timing or scalar expectancy theory Gibbon 1977 is a model that posits an internal clock , and particular memory and Decision making decision processes. This is one of the most popular views of timing in animals. The clock and memory are driven by a discrete cardiac pacemaker pacemaker accumulator mechanism that yields a linear Scale ratio scale for encoded time. The scalar expectancy theory SET posits that animals make choices based on a single sample. The animals are posited to make estimates of the time to reinforcement delivery using a scalar timing process. This scalar timing process rescales estimates for different values of the interval being timed. Scalar timing implies a constant coefficient of variation. Expectations or reinforcement are based on these estimates are formed from these sample. The animal discriminates between response alternatives by taking the ratio of their expectancies. A number of alternatives have been developed over the years. These include Killeen s 1991 Behavioral Theory of timing BeT model and Machado s 2005 learning to time LeT model. References Gibbon, J. 1977 . Scalar expectancy theory and Weber s law in animal timing. Psychological Review . 84 3 , 279 325 Kacelnik, A., & Brunner, D. 2002 . Timing and foraging Gibbon s scalar expectancy theory and optimal patch exploitation. Learning and Motivation 33 1 , 177 195. Killeen, P. R. 1991 . Behavior s time. In G. Bower Ed. , The psychology of learning and motivation Vol. 27, pp. 294 334 . New York Academic Press Machado, A., & Keen, R. 1999 . Learning to Time LET or Scalar Expectancy Theory SET ? A critical test of two models of timing. Psychological Science , 10, 285 290. Machado, A., & Pata, P. 2005 . Testing the Scalar Expectancy Theory SET and the Learning to Time model LeT in a double bisection task. Behavior and Learning , 33, 111 122. Category Theories Category Time ... more details
Scalar processors represent the simplest class of computer processor s. A scalar processor processes one datum at a time typical data items being integer computer science integer s or floating point number s . ref name ram Advanced Microprocessors and Interfacing by Badri Ram 2000 ISBN 0070434484 page 11 ref , a scalar processor is classified as a SISD processor Single Instructions, Single Data . In a vector processor , by contrast, a single instruction operates simultaneously on multiple data items. called SIMD in Flynn s taxonomy The difference is analogous to the difference between scalar mathematics scalar and Vector geometric vector arithmetic. A superscalar processor, on the other hand, executes more than one instruction during a clock cycle by simultaneously dispatching multiple instructions to redundant functional units on the processor. Each functional unit is not a separate CPU core but an execution resource within a single CPU such as an arithmetic logic unit, a bit shifter, or a multiplier. ref name ram See also Instruction pipeline Parallel computing Superscalar References Reflist Compu hardware stub DEFAULTSORT Scalar Processor Category Central processing unit cs Skal rn procesor de Skalarprozessor es Procesador escalar fr Processeur scalaire ko id Prosesor skalar mk ja pl Procesor skalarny ru uk zh ... more details
Dablink See scalar disambiguation scalar for an account of the broader concept also used in physics and computing ... space through the operation of scalar multiplication , in which a vector can be multiplied by a number ... of the associated field. A inner product scalar product operation not to be confused with scalar multiplication may be defined on a vector space, allowing two vectors to be multiplied to produce a scalar. A vector space equipped with a scalar product is called an inner product space . The real component of a quaternion is also called its scalar part . The term is also sometimes used ... and an n × 1 matrix, which is formally a 1× 1 matrix, is often said to be a scalar . The term scalar matrix is used to denote a matrix of the form kI where k is a scalar and I is the identity matrix . Etymology The word scalar derives from the Latin word scalaris , adjectival form from scala ... recorded usage of the word scalar in mathematics was by Fran ois Vi te in Analytic Art In artem analyticen ... from one kind to another are called scalar terms. According to a citation in the Oxford ... of numbers from negative to positive infinity we shall call it therefore the scalar part. Definitions ... of scalars, and a scalar multiplication operation that takes a scalar k and a vector v to another vector k v . For example, in a coordinate space , the scalar multiplication math k v 1, v 2, dots, v ... algebra basis . It follows that every vector space over a scalar field K is isomorphism isomorphic ... norm function that assigns to every vector v in V a scalar v . By definition, multiplying v by a scalar ... nowiki s scalar field K , which restricts the latter to fields that support the notion of sign. Moreover ..., not every scalar product space is a normed vector space. Scalars in modules When the requirement ... forms a module over the algebra of real functions on the manifold. Scaling transformation The scalar ... of linear transformation . Scalar operations computer science Operations that apply to a single value ... more details
In mathematics , scalar multiplication is one of the basic operations defining a vector space in linear algebra ref cite book last Lay first David C. title Linear Algebra and Its Applications publisher Addison Wesley year 2006 edition 3rd isbn 0 321 28713 4 ref ref cite book last Strang first Gilbert authorlink Gilbert Strang title Linear Algebra and Its Applications publisher Brooks Cole year 2006 edition 4th isbn 0 03 010567 6 ref ref cite book last Axler first Sheldon title Linear Algebra Done Right publisher Springer Science Business Media Springer year 2002 edition 2nd isbn 0 387 98258 2 ref or more generally, a module mathematics module in abstract algebra ref cite book last1 Dummit first1 David S. last2 Foote first2 Richard M. title Abstract Algebra publisher John Wiley & Sons year 2004 ... 2002 isbn 0 387 95385 X ref . In an intuitive geometrical context, scalar multiplication of a real ... changing its direction. The term scalar itself derives from this usage a scalar is that which scales vectors. Scalar multiplication is different from the scalar product , which is an inner product ... over K , then scalar multiplication is a function mathematics function from K × V to V . The result of applying this function to c in K and v in V is denoted c v . Scalar multiplication obeys the following ... in either. Juxtaposition indicates either scalar multiplication or the multiplication operation in the field. Scalar multiplication may be viewed as an external mathematics external binary operation or as an group action action of the field on the vector space. A geometric interpretation to scalar ... itself and scalar multiplication may then be taken to be simply the multiplication in the field. When V is K sup n sup , then scalar multiplication is defined component wise . The same idea goes ... left scalar multiplication c v and right scalar multiplication v c . See also Statics Mechanics Product mathematics References reflist DEFAULTSORT Scalar Multiplication Category Linear algebra ... more details
Image Dot Product.svg thumb 300px right If 0 90 , as in this case, the scalar projection of a on b coincides with the Euclidean norm length of the vector projection . File Projection and rejection.png thumb 200px Vector projection of a on b a sub 1 sub , and vector rejection of a from b a sub 2 sub . The scalar projection of a vector geometric vector math mathbf a math on or onto a vector math mathbf b math , also known as the scalar resolute or scalar component of math mathbf a math in the direction of math mathbf b math , is given by math s mathbf a cos theta mathbf a cdot mathbf hat b , math where the operator math cdot math denotes a dot product , math hat mathbf b math is the unit vector in the direction of math mathbf b math , math mathbf a math is the Euclidean norm length of math mathbf a math , and math theta math is the angle between math mathbf a math and math mathbf b math . The scalar projection is a scalar mathematics scalar , equal to the Euclidean norm length of the orthogonal projection of math mathbf a math on math mathbf b math , with a minus sign if the projection has an opposite direction with respect to math mathbf b math . Multiplying the scalar projection of math mathbf a math on math mathbf b math by math mathbf hat b math converts it into the above mentioned orthogonal projection, also called vector projection of math mathbf a math on math mathbf b math . Definition based on angle If the angle math theta math between math mathbf a math and math mathbf b math is known, the scalar projection of math mathbf a math on math mathbf b math can be computed using math s mathbf a cos theta . math Definition in terms of a and b When math theta math is not known ... a cdot mathbf b mathbf a , mathbf b cos theta , math By this property, the definition of the scalar ... b mathbf a , mathbf b frac mathbf a cdot mathbf b mathbf b , math Properties The scalar projection has .... See also Scalar product Cross product Vector projection Category Vectors Mathanalysis stub ca Projecci ... more details
In high energy physics , a scalar meson is a meson with total angular momentum quantum number total spin 0 and even parity physics parity usually noted as J sup P sup 0 sup sup . Compare to pseudoscalar meson . These mesons are most often observed in proton antiproton annihilation, radiative decays of vector mesons , and meson meson scattering. The first known scalar mesons have been observed since .... The light unflavored scalar mesons may be divided into three groups those having a mass below ... radially excited unflavored scalar mesons above 2 GeV c sup 2 sup . The heavier scalar mesons containing ... made to determine the quark content of the lighter scalar mesons however, no consensus has yet been reached. The scalar mesons in the mass range of 1 GeV c sup 2 sup to 2 GeV c sup 2 sup are generally .... E. Close, An Introduction to Quarks and Partons , Academic Press New York 1979 , pgs. 88 89 ref The scalar ... decay characteristics. The scalar mesons in the mass range below 1 GeV c sup 2 sup are much more ... scalar mesons were often interpreted within the framework of the linear sigma model , and many theorists still choose this interpretation of the scalar mesons as the chiral partners of the pseudoscalar .... D15, 267 1977 ref the lightest scalar mesons have been interpreted by some theorists to be possible ... Press New York 1986 , Vol. II pgs. 409 419 ref where the scalar tetraquarks are actually predicted to have lower mass than the conventional scalar mesons. This picture of the scalar mesons seems to fit ... of the unflavored scalar mesons began with the Crystal Ball and Crystal Barrel experiments of the mid ... introduction of the sigma meson as an acceptable candidate for a light scalar meson in 1996 by Tornqvist ... into the lightest scalar mesons were conducted with renewed interest. The http pdg.lbl.gov Particle ... the scalar mesons. Examples confirmed K sub 0 sub sup sup 1430 candidates K sub 0 sub sup sup 800 ... , f sub 0 sub 1790, X 1810 References references DEFAULTSORT Scalar Meson Category Mesons ko sl ... more details
Unreferenced date December 2009 In Riemannian geometry , the scalar curvature or Ricci scalar is the simplest ..., the scalar curvature represents the amount by which the volume of a geodesic ball in a curved ..., the scalar curvature is twice the Gaussian curvature , and completely characterizes the curvature ... more than one functionally independent quantity. In general relativity , the scalar curvature is the Lagrangian ... metrics are known as Einstein manifold Einstein metrics . The scalar curvature is defined as the trace ... involving only the scalar curvature are extremely subtle and difficult. One of the few is the positive ... , which seeks extremal metrics in a given conformal class for which the scalar curvature is constant. Definition The scalar curvature is usually denoted by S other notations are Sc , R . It is defined ... operatorname Ric R ij ,dx i otimes dx j. math Given a coordinate system and a metric tensor, scalar ... both can be naturally defined for any affine connection , the scalar curvature requires a metric ... Finsler geometry . Direct geometric interpretation When the scalar curvature is positive at a point ... in Euclidean space. On the other hand, when the scalar curvature is negative at a point, the volume ..., in order to characterize the precise value of the scalar curvature S at a point p of a Riemannian ... at radius 0, is exactly minus the scalar curvature divided by 3 n 2 . Boundaries ... 4 . math Special cases Surfaces In two dimensions, scalar curvature is exactly twice the Gaussian curvature ..., scalar curvature of a sphere with radius r is equal to 2 r sup 2 sup . The 2 dimensional Riemann tensor has only one independent component and it can be easily expressed in terms of the scalar curvature ... tensor of an n dimensional Euclidean space vanishes identically, so the scalar curvature ... 1 r sup 2 sup . Hence the scalar curvature is S n n &minus 1 r sup 2 sup . Hyperbolic space ... &minus 1 r sup 2 sup . The scalar curvature is thus S &minus n n &minus 1 r sup 2 sup . Traditional ... more details
Unreferenced date December 2009 In the theory of pseudo Riemannian manifold Lorentzian manifolds , particularly in the context of applications to general relativity , the Kretschmann scalar is a quadratic curvature invariant general relativity scalar invariant . It was introduced by Erich Kretschmann . Definition The Kretschmann invariant is math K R abcd , R abcd math where math R abcd math is the Riemann curvature tensor . Because it is a sum of squares of tensor components, this is a quadratic invariant. Relation to other invariants Another possible invariant which has been employed for example in writing the gravitational term of the Lagrangian for some higher order gravity theories is math C abcd , C abcd math where math C abcd math is the Weyl tensor , the conformal curvature tensor which is also the completely traceless part of the Riemann tensor. In math d math dimensions this is related to the Kretschmann invariant by math R abcd , R abcd C abcd , C abcd frac 4 d 2 R ab , R ab frac 2 d 1 d 2 R 2 math where math R ab math is the Ricci curvature tensor and math R math is the Ricci scalar curvature obtained by taking successive traces of the Riemann tensor . The Kretschmann scalar and the Chern Pontryagin scalar math R abcd , star R abcd math where math star R abcd math is the left dual of the Riemann tensor, are mathematically analogous to some extent, physically analogous to the familiar invariants of the electromagnetic field tensor math F ab , F ab , F ab , star F ab math See also Carminati McLenaghan invariants , for a set of invariants. Classification of electromagnetic fields , for more about the invariants of the electromagnetic field tensor. Curvature invariant , for curvature invariants in Riemannian and pseudo Riemannian geometry in general. Curvature invariant general relativity . Ricci decomposition , for more about the Riemann and Weyl tensor. DEFAULTSORT Kretschmann Scalar Category Riemannian geometry Category Lorentzian manifolds Category Tenso ... more details
In pragmatics , scalar implicature , or quantity implicature , ref cite journal last Hansen first Maj Britt Mosegaard coauthors Erling Strudsholm date May 1, 2008 journal Linguistics An Interdisciplinary Journal of the Language Sciences title The semantics of particles advantages of a contrastive and panchronic approach a study of the polysemy of French deja and Italian gia. volume 46 issue 3 pages 471 publisher Walter de Gruyter quote Moreover, the truth of a sentence like It s a pretty big thing in itself if I recover the outlay. with an inherently scalar predicate allows, in principle, for the truthful ... it does not logically imply it that Bill does not have all his money in cash. Origin Scalar implicatures ... ti mesi halfway . Such expressions are known to give rise to scalar inferences cross linguistically ... limited success in deriving scalar implicatures from the use of aspectual verbs but they succeed with discrete ... limited success in deriving scalar implicatures from the use of aspectual verb s such as start which ... children succeed in deriving scalar implicatures with discrete degree modifiers such as half as in half finished. ref name insights Their ability to spontaneously compute scalar implicatures was greater than their ability to judge the pragmatic appropriateness of scalar statements. ref name insights In addition, the tested children were able to suspend scalar implicatures in environments where .... Examples of scalar implicature Some examples of scalar implicature ref taken from Carston ref are 1a ..., Relevance and Scalar Implicature http www.phon.ucl.ac.uk home robyn pdf scalarinference.pdf ... and Non detachability , http www.sc.ehu.es ylwkocak papers VSZ.report.pdf Angelika Kratzer, Scalar Implicatures Are There Any? Workshop on Polarity, Scalar Phenomena, and Implicatures. University of Milan Bicocca June 18, 2003 http semanticsarchive.net Archive GJiYjQxN Scalar 20Implicatures.pdf Ira Noveck, When children are more logical than adults experimental investigations of scalar implicature ... more details
In mathematics, a relative scalar of weight w is a scalar valued function whose transform under a coordinate transform, math bar x j bar x j x i math on an n dimensional manifold obeys the following equation math bar f bar x j J w f x i math where math J begin vmatrix displaystyle frac partial x 1, ldots,x n partial bar x 1, ldots, bar x n end vmatrix , math that is, the determinant of the Jacobian of the transformation. ref name lovelock cite book last Lovelock first David title Tensors, Differential Forms, and Variational Principles date April 1, 1989 publisher Dover isbn 0486658406 pages 400 url http store.doverpublications.com 0486658406.html coauthors Rund accessdate 19 April 2011 page 103 format Paperback chapter 4 ref Relative scalars are an important special case of the more general concept of a relative tensor . Ordinary scalar An ordinary scalar or absolute scalar ref cite book last Veblen first Oswald title Invariants of Quadratic Differential Forms year 2004 publisher Cambridge University Press isbn 0521604842 pages 112 page 21 ref refers to the math w 0 math case. If math x i math and math bar x j math refer to the same point math P math on the manifold, then we desire math bar f bar x j f x i math . This equation can be interpreted two ways when math bar x j math are viewed as the new coordinates and math x i math are viewed as the original coordinates . The first is as math bar f bar x j f x i bar x j math , which converts the function to the new coordinates . The second is as math f x i bar f bar x j x i math , which converts back to the original coordinates. Of course, new or original is a relative concept. There are many physical quantities that are represented ... of an ordinary scalar depends on the coordinate system used. This coordinate dependence tends to remove any physical meaning from the integral of an ordinary scalar. Scalar density A scalar density .... It can be shown the determinant of a type 0,2 tensor is a relative scalar of weight 2. Generalization ... more details
the scalar potential of electromagnetism electric potential all other uses potential File Mass ... mathbf F nabla P math A scalar potential is a fundamental concept in vector analysis and physics the adjective scalar is frequently omitted if there is no danger of confusion with vector potential . The scalar potential is an example of a scalar field . Given a vector field F , the scalar potential ... per unit length. In order for F to be described in terms of a scalar potential only, the following ... field that is a gradient of a differentiable single valued function single valued scalar field P. The second condition is a requirement of F so that it can be expressed as the gradient of a scalar ... is said to be irrotational vector field irrotational Conservative . Scalar potentials play a prominent role in many areas of physics and engineering. The gravity potential is the scalar ... the electric potential is the scalar potential associated with the electric field , i.e. ... field s have a scalar potential only in the special case when it is a Laplacian field . Certain aspects ... mechanics . Further, the scalar potential is the fundamental quantity in quantum mechanics . Not every vector field has a scalar potential. Those that do are called conservative vector field conservative ... in terms of a scalar potential and corresponding vector potential . In electrodynamics the electromagnetic scalar and vector potentials are known together as the electromagnetic four potential ..., then math mathbf F nabla V, math so that V is a scalar potential of the conservative vector field F . Scalar potential is not determined by the vector field alone indeed, the gradient of a function ... it. This creates enough buoyant force to counteract gravity. Calculating the scalar potential Given a vector field E , its scalar potential can be calculated to be math phi mathbf R 0 1 over 4 pi ... potential Yukawa potential References references DEFAULTSORT Scalar Potential Category Potential ... more details
Unreferenced date December 2007 In theoretical physics , scalar electrodynamics is a theory of a U 1 gauge field coupled to a charged spin 0 scalar field that takes the place of the Dirac fermion s in ordinary quantum electrodynamics . The scalar field is charged, and with an appropriate potential, it has the capacity to break the gauge symmetry via the Abelian Higgs mechanism . The model consists of a complex scalar field math phi x math minimally coupled to a gauge field math A mu x math . The dynamics is given by the Lagrangian density math mathcal L frac12 D mu phi D mu phi U phi phi frac14 F mu nu F mu nu , math where math F mu nu partial mu A nu partial nu A mu math is the electromagnetic field strength, math D mu phi partial mu phi i e A mu phi math is the covariant derivative of the field math phi math , math e math is the electric charge and math U phi phi math is the potential for the complex scalar field. This model is invariant under gauge transformations parametrized by math lambda x math math phi x e ie lambda x phi x quad textrm and quad A mu x A mu x partial mu lambda x . math If the potential is such that its minimum occurs at non zero value of math phi math , this model exhibits the Higgs mechanism . This can be seen by studying fluctuations about the lowest energy configuration, one sees that gauge field behaves as a massive field with its mass proportional to the math e math times the minimum value of math phi math . As shown in 1973 by Nielsen and Olesen, this model, in math 2 1 math dimensions, admits time independent finite energy configurations corresponding to vortices carrying magnetic flux. The magnetic flux carried by these vortices are quantized in units of math tfrac 2 pi e math and appears as a topological charge associated with the topological current math J top mu epsilon mu nu rho F nu rho . math These vortices are similar to the vortices appearing in type II superconductors. This analogy was used by Nielsen and Olesen in obtaining their solutions ... more details
In physics , a Lorentz scalar is a scalar physics scalar which is invariant under a Lorentz transformation . A Lorentz scalar may be generated from multiplication of vectors or tensors. While the components of vectors and tensors are in general altered by Lorentz transformations, scalars remain unchanged. Simple scalars in special relativity The length of a position vector Image Fermi walker 1.png frame left World lines for two particles at different speeds. In special relativity the location of a particle in 4 dimensional spacetime is given by its world line math x mu ct, mathbf x math where math mathbf x mathbf v t math is the position in 3 dimensional space of the particle, math mathbf v math is the velocity in 3 dimensional space and math c math is the speed of light . The length of the vector is a Lorentz scalar and is given by math x mu x mu eta mu nu x mu x nu ct 2 mathbf x cdot mathbf x stackrel mathrm def tau 2 math where math tau math is c times the proper time as measured by a clock in the rest frame of the particle and the metric is given by math eta mu nu eta mu nu begin pmatrix 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 0 & 0 & 0 & 1 end pmatrix math . This is a time like metric. Often the Minkowski metric is used in which the signs of the ones are reversed. math eta mu nu eta mu nu begin pmatrix 1 & 0 & 0 & 0 0 & 1 & 0 & 0 0 & 0 & 1 & 0 0 & 0 & 0 & 1 end pmatrix math . This is a space like metric. In the Minkowski metric the space like interval math s math is defined ... scalar and is minus one, math v mu v mu 1 , math . The 4 velocity is therefore, not only a representation ... product of the acceleration and the velocity is a Lorentz scalar and is zero. This rotation is simply ... particle in the frame of the second particle, is a Lorentz scalar. Therefore math E 1 over c 2 gamma ... is math p mu p mu m 2 , math . Therefore math m 2 math is a Lorentz scalar. The relationship remains ... particle is a Lorentz scalar. Measurement of the 3 speed of the particle The 3 speed, in the frame of the second ... more details
In General Relativity , the Weyl scalars are a set of five complex Scalar physics scalar quantities , math Psi 0, ldots, Psi 4 math , describing the curvature of a four dimensional spacetime . They are the expression of the ten independent degrees of freedom of the Weyl tensor math C abcd math in the Newman Penrose Formalism for general relativity . Given a null tetrad math l a, n a, m a, bar m a math , the scalars are given up to an overall conventional sign by math Psi 0 C alpha beta gamma delta l alpha m beta l gamma m delta , math math Psi 1 C alpha beta gamma delta l alpha n beta l gamma m delta , math math Psi 2 C alpha beta gamma delta l alpha m beta bar m gamma n delta , math math Psi 3 C alpha beta gamma delta l alpha n beta bar m gamma n delta , math math Psi 4 C alpha beta gamma delta n alpha bar m beta n gamma bar m delta . math Physical Interpretation Szekeres 1965 ref cite journal author P. Szekeres title The Gravitational Compass journal Journal of Mathematical Physics year 1965 volume 6 issue 9 pages 1387 1391 doi 10.1063 1.1704788 bibcode 1965JMP.....6.1387S . ref gave an interpretation of the different Weyl scalars at large distances math Psi 2 math is a Coulomb term, representing the gravitational monopole of the source math Psi 1 math & math Psi 3 math are ingoing and outgoing longitudinal radiation terms math Psi 0 math & math Psi 4 math are ingoing and outgoing transverse radiation terms. For a general asymptotically flat spacetime containing radiation Petrov Type I , math Psi 1 math & math Psi 3 math can be transformed to zero by an appropriate choice of null tetrad. Thus these can be viewed as gauge quantities. A particularly important case is the Weyl scalar math Psi 4 math . It can be shown to describe outgoing gravitational radiation in an asymptotically flat spacetime as math Psi 4 frac 1 2 left ddot h hat theta hat theta ddot h hat phi hat phi right i ddot h hat theta hat phi ddot h i ddot h times . math Here, math h math and math h times ... more details
In Riemannian geometry , a branch of mathematics , the prescribed scalar curvature problem is as follows given a closed manifold closed , smooth manifold M and a smooth, real valued function &fnof on M , construct a Riemannian metric on M whose scalar curvature equals &fnof . Due primarily to the work of Jerry Kazdan J. Kazdan and F. Warner in the 1970s, this problem is well understood. The solution in higher dimensions If the dimension of M is three or greater, then any smooth function &fnof which takes on a negative value somewhere is the scalar curvature of some Riemannian metric. The assumption that &fnof be negative somewhere is needed in general, since not all manifolds admit metrics which have strictly positive scalar curvature. For example, the three dimensional torus is such a manifold. However, Kazdan and Warner proved that if M does admit some metric with strictly positive scalar curvature, then any smooth function &fnof is the scalar curvature of some Riemannian metric. See also Prescribed Ricci curvature problem Yamabe problem References Aubin, Thierry. Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics, 1998. Kazdan, J., and Warner F. Scalar curvature and conformal deformation of Riemannian structure. Journal of Differential Geometry. 10 1975 . 113&ndash 134. Category Riemannian geometry Category Mathematical problems Category Curvature mathematics Scalar curvature differential geometry stub ... more details
Scalar theories of gravitation are field theories of gravitation in which the gravitational field is described using a Scalar field theory scalar field , which is required to satisfy some field equation ... field. Newtonian gravity The prototypical scalar theory of gravitation is Newtonian gravity Newtonian ... The first attempts to present a relativistic classical field theory of gravitation were also scalar ... A math is a scalar which is a function of position. This suggestion signifies that the inertial mass should depend on the scalar field. Nordstr m s second theory satisfies the weak equivalence ... relativity. Einstein s scalar theory In 1913, Einstein erroneously concluded from his hole argument that general covariance was not viable. Inspired by Nordstr m s work, he proposed his own scalar theory. This theory employs a massless scalar field coupled to the stress energy tensor, which is the sum ... of the scalar field itself. The second represents the stress momentum energy of any matter which ... consistency condition, so Einstein dropped this theory in late 1914. Associating the scalar field ... not be a scalar theory. Indeed, the theory he finally arrived at in 1915, general relativity , is a tensor theory, not a scalar theory, with a 2 tensor, the metric, as the potential. Unlike his 1913 scalar theory, it is generally covariant , and it does take into account the field energy momentum ... math k mu k nu math , the source of scalar gravitational fields come from the trace of this which is equal to the square of the rest mass of a particle just like Higgs fields. Thus the theory of scalar .... Additional variations Kaluza Klein theory involves the use of a scalar gravitational field in addition ... Dicke theory is a scalar tensor theory, not a scalar theory, meaning that it represents the gravitational interaction using both a scalar field and a tensor field. We mention it here because one of the field equations of this theory involves only the scalar field and the trace of the stress energy ... more details
Expert subject Mathematics date November 2008 In theoretical physics , a scalar tensor theory is a theory that includes both a Scalar field theory scalar field and a tensor field to represent a certain interaction. For example, the Brans Dicke theory of gravitation uses both a scalar field and a tensor ... theories to explain reality. These fields can be scalar, vectorial or tensorial. For them, there is Scalars ... a direction. In them, every component of the direction is a scalar. Tensors degree 2 are a wider ... of scalar quantities named potentials. In classical physics before Albert Einstein Einstein , gravitation ... a scalar potential field, dependent of the mass of the particles. Thus, Isaac Newton Newtonian gravity is called a scalar theory . The gravitational force is dependent of the distance r of the massive ... in this context is by using both tensor of degree n 1 and scalar fields, i.e. so that gravitation is not only given through a scalar field nor through the metric. These are scalar tensor theories of gravitation .... It is a scalar and gauge invariant look at gauge theories quantity dependent on the curvature scalar ... with a square scalar field, field theories of scalar tensor theories of gravitation ... dependent of the scalar field. Higher dimensional relativity and scalar tensor theories After ... of the scalar field, to change energy conservation as well, according to the ideas of Paul Dirac Dirac ... to theories of usual General Relativity in 4 dimensions with an additional scalar field. One ..., dependent of the mass distribution in the universe, as a function of a scalar variable, coupled as a field in the Lagrangian. It uses a scalar field of infinite length scale i.e. long ranged , so, in the language of Yukawa s theory of nuclear physics, this scalar field is a massless field . This theory becomes Einsteinian for high values for the parameter of the scalar field. In 1979, R. Wagoner proposed a generalization of scalar tensor theories using more than one scalar field coupled to the scalar ... more details
In general relativity , a scalar field solution is an Exact solutions in general relativity exact solution of the Einstein field equation in which the gravitational field is due entirely to the field energy and momentum of a scalar field . Such a field may or may not be massless , and it may be taken to have minimal curvature coupling , or some other choice, such as conformal coupling . Mathematical ... of the gravitational field . In addition, we must specify a scalar field by giving a function math ... match the energy momentum density stress energy tensor for the scalar field, which in the simplest case, a minimally coupled massless scalar field , can be written math G ab 8 pi left psi a psi b frac ... densities in field theory Lagrangian density for the scalar field, which in the case of a minimally coupled massless scalar field is math L g mn , psi m , psi n math Here, math frac delta ... equation in the case where the field energy of the scalar field is the only source of the gravitational field . Physical interpretation Scalar fields are often interpreted as classical approximations ... quintessence physics quintessence field can appear as a scalar field. For example, a flux of neutral pion s can in principle be modeled as a minimally coupled massless scalar field. Einstein ... coupled massless scalar field , an adapted frame math vec e 0, vec e 1, vec e 2, vec e 3 math the first ... of the scalar field. Eigenvalues The characteristic polynomial of the Einstein tensor in a minimally coupled massless scalar field solution must have the form math chi lambda lambda 8 pi sigma ... 3 4 math math G a b , G b c , G c d , G d a R 4 4 math Examples Notable individual scalar field solutions include the Janis Newman Winicour scalar field solution , which is the unique static and spherically symmetric massless minimally coupled scalar field solution. See also Exact solutions in general ... coupled scalar field. Category Exact solutions in general relativity ... more details
by baryonic matter ordinary matter alone. Scalar field The dark matter can be modeled as a scalar ... with a mass of 1.1 10 sup 23 sup eV ref T. Matos and L. A. Ure a L pez, Quintessence and Scalar ... preprint A Further Analysis of a Cosmological Model of Quintessence and Scalar Dark Matter , Phys ..., T. Matos, D. N ez, L. A. Ure a L pez and P. Wiederhold, Galactic Collapse of Scalar Field Dark .... A. Ure a L pez, Evolution of the Schr dinger Newton system for a self gravitating scalar field ... A. Bernal and F. S. Guzm n, Scalar Field Dark Matter non spherical collapse and late time behavior ... more details
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